The younger flagellum sets the beat for Chlamydomonas reinhardtii

Eukaryotes swim with coordinated flagellar (ciliary) beating and steer by fine-tuning the coordination. The model organism for studying flagellate motility, Chlamydomonas reinhardtii, employs synchronous, breaststroke-like flagellar beating to swim, and it modulates the beating amplitudes differentially to steer. This strategy hinges on both inherent flagellar asymmetries (e.g. different response to chemical messengers) and such asymmetries being effectively coordinated in the synchronous beating. In C. reinhardtii, the synchrony of beating is known to be supported by a mechanical connection between flagella; however, how flagellar asymmetries persist in the synchrony remains elusive. For example, it has been speculated for decades that one flagellum leads the beating, as its dynamic properties (i.e. frequency, waveform, etc.) appear to be copied by the other one. In this study, we combine experiments, computations, and modeling efforts to elucidate the roles played by each flagellum in synchronous beating. With a non-invasive technique to selectively load each flagellum, we show that the coordinated beating essentially only responds to load exerted on the cis flagellum; and that such asymmetry in response derives from a unilateral coupling between the two flagella. Our results highlight a distinct role for each flagellum in coordination and have implication for biflagellates’ tactic behaviors.


Introduction
The ability to swim towards desirable environments and away from hazardous ones is fundamental to the survival of many microorganisms.These so-called tactic behaviors are exhibited by many motile microorganisms ranging from bacteria (1,2) to larger flagellates and ciliates (3)(4)(5).Different microorganisms have developed specific strategies for steering, depending on the tactic behavior and on their specific sensory and motility repertoire.For example, bacteria modulate the tumbling rate (1) while flagellates and ciliates modulate the waveform (6)(7)(8)(9), amplitude (10,11) and frequency of their flagellar/ciliary (4,12) beating.The goal of these active modulations of the motility is to achieve a spatially asymmetric generation of propulsive force to steer the organism.
C. reinhardtii (CR), the model organism for studies of flagellar motility, achieves tactic navigation by a fine-tuned differential modulation on its two flagella.Studying this organism offers great opportunities to look into how flagella coordinate with each other and how such coordination helps facilitate targeted steering.CR has a symmetric cell body and two near-identical flagella inherited from the common ancestors of land plants and animals (13).It swims by beating its two flagella synchronously and is capable of photo-and chemotaxis (10,14).For this biflagellated organism, effective steering hinges on both flagellar asymmetry and flagellar coordination.On the one hand, the two flagella must be asymmetric to respond differentially to stimuli (10,15); on the other hand, the differential responses must be coordinated by the cell such that the beating would remain synchronized to guarantee effective swimming.Understanding this remarkable feat requires knowledge about both flagellar asymmetry and coordination.
The two flagella are known to be asymmetric in several, possibly associated, aspects.First of all, they differ in developmental age (16,17).The flagellum closer to the eyespot, the cis(eyespot) flagellum, is always younger than the other one, the trans(-eyespot) flagellum.This is because the cis is organized by a basal body (BB) that develops from a pre-matured one in the mother cell; and this younger BB also organizes the flagellar root (D4 rootlet) that dictates the eyespot formation (18).Second, the two flagella have asymmetric protein composition (19)(20)(21).For example, the trans flagellum is richer in CAH6, a protein possibly involved in CO 2 sensing (14,20).Finally, the flagella have different dynamic properties (22)(23)(24).Their beating is modulated differentially by second messengers such as calcium (22,23) and cAMP (25).When beating alone, the trans beats at a frequency 30%-40% higher than the cis (23,(26)(27)(28); the trans also displays an attenuated waveform (29) and a much stronger noise (29,30).
Remarkably, despite these inherent asymmetries, CR cells establish robust synchronization between the flagella.Such coordination enables efficient swimming and steering of the cells and takes basis on the fibrous connections between flagellar bases (31,32).Intriguingly, in the coordinated beating, both flagella display dynamic properties, i.e., flagellar waveform, beating frequency (∼50 Hz), and frequency fluctuation, that are more similar to those of the cis flagellum (26,(28)(29)(30)33).This has led to a long-standing hypothesis that "the cis somehow tunes the trans flagellum" (26).This implies that the symmetric flagellar beating ("breast-stroke") observed is the result of interactions between two flagella playing differential roles in coordination.How does the basal coupling make this possible?Recent theoretical efforts show that the basal coupling can give rise to different synchronization modes (34)(35)(36); and that flagellar dynamics, such as beating frequency, may simply emerge from the interplay between mechanics of basal coupling and bio-activity (36).Yet, most theoretical efforts examining flagellar synchronization have assumed two identical flagella, limiting the results' implication for the realistic case.Moreover, little experiments directly probe the flagella's differential roles during synchronous beating (37).Therefore, flagellar coordination in this model organism remains unclear.To clarify the picture experimentally, one needs to selectively force each flagellum, and characterize the dynamics of the flagellar response.
In this study, we address this challenge and devise a non-invasive approach to apply external forces selectively on the cis-or the trans-flagella.Oscillatory background flows are imposed along an angle with respect to the cell's symmetry axis.Such flows result in controlled hydrodynamic forces, which are markedly different on the two flagella.With experiments, hydrodynamic computations, and modeling, we show definitively that the two flagella are unilaterally coupled, such that the younger flagellum (cis) coordinates the beating, whereas the elder one simply copies the dynamic properties of the younger.This also means that only external forces on the cis may mechanically fine-tune the coordination.We also study the effect of calcium in the cis' leading role as calcium is deeply involved in flagellar asymmetry and hence phototactic steering.In addition, a well-known mutant that lacks flagellar dominance (ptx1) (23,38) is examined.Results show that the coordinating role of cis does not need environmental free calcium, whereas it does require the genes lost in ptx1.Our results discern the differential roles of CR's flagella, highlight an advanced function of the inter-flagellar mechanical coupling, and have implications for biflagellates' tactic motility.

Experimental scheme for selective loading
We set out to establish a non-invasive experimental technique that exerts differential loads on the flagella of CR.Following the study by Quaranta et  For given realistic flagellar shapes, we compare computed loads with and without external flows.
From these we isolate the loads from the induced flows F Flow and P Flow (Methods).
Loads on each flagellum under flows of θ = 0 • , −45 • , 45 • are presented in Fig. 2. Upper panels display the magnitude of the drag force F Flow = |F Flow |; while lower panels show viscous power P Flow .Force magnitudes are scaled by F 0 = 6πµRU 0 = 9.9 pN; while the powers by P 0 = F 0 U 0 = 1.1 fW.F 0 is the Stokes drag on a typical free-swimming cell (radius R = 5 µm, speed U 0 = 110 µm/s, water viscosity µ = 0.95 mPa•s).
Evidently, along θ = 0 • , flows load the flagella equally (Fig. 2A).However, at θ = −45 • , flows load the cis flagellum ∼ 2 times larger than the trans (Fig. 2B, F c Flow ≈ 2F t Flow ); whereas flows at θ = 45 • do the opposite (Fig. 2C).The selectivity also manifests in (the absolute values of) P Flow .We do notice that flows along θ = +45 • are able to synchronize the flagella with P Flow < 0, meaning that the flagella are working against the flows, and this shall be discussed in later sections.
Hereon forward, we refer to θ c -flows, flows for which θ = −45 • and the cis-flagellum is selectively loaded.Likewise, θ t -flows denote flows on θ = +45 • that selectively load the trans.θ a -flows denote the axial flow along θ = 0 • .We next introduce how we quantify the flows' effective forcing strength (ε) on the cell.
Phase dynamics of flagellar beating is extracted from videography (31,39,40).Recordings are masked and thresholded to highlight the flagella (Fig. 1B-C).Then the mean pixel values over time within two sampling windows (Fig. 1D) are converted to observable-invariant flagellar phases (41), Fig. 1E.Throughout this study, as cis and trans always beat synchronously (Fig. 1E inset), their phases ϕ c,t are used interchangeably as the flagellar phase ϕ.The flagellar phase dynamics under external periodic forcing is described by Adler equation (42)(43)(44): ∆ϕ = ϕ − 2πf f t is the phase difference between the beating and the forcing, with f f the forcing frequency, and ε the forcing strength.The detuning ν = f f − f 0 is the frequency mismatch between the beating (f 0 ) and forcing.ζ(t) represents a white noise that satisfies , with T eff an effective temperature and δ(t) the Dirac delta function.
When the forcing strength outweighs the detuning (ε > |ν|), synchronization with the flow (d∆ϕ/dt = 0) emerges, see the plateaus marked black in Fig. 1F.We characterize synchronization with τ = t sync /t tot , where t sync is the total time of flow synchronization and t tot the flow duration.Fig. 1F presents the phase dynamics which are representative and range from: no synchronization (τ =0, i), unstable synchronization (0 < τ < 1, ii-iii), and stable synchronization (τ =1, iv).In this study, the frequency range in ν for which τ ≥ 0.5 is used to measure ε (see Fig. 1F inset).This method is equivalent to previous fitting-based methods (28,31), see SM. Sec.S1.

Asymmetric susceptibility to flow synchronization
Now we examine cell responses to flows of various amplitudes and along different directions.First we explore flow synchronization over a broad range of amplitudes and frequencies.θ aflows with frequencies f f ∈ [40,75] Hz and amplitudes U ∈ [390, 2340] µm/s are imposed.The scanned range covers reported intrinsic frequencies of both the cis and trans flagellum (22,24,26,27); while the amplitude reaches the maximum instantaneous speed of a beating flagellum (∼ 2000 µm/s).Fig. 3A displays the resultant flow-synchronized time fractions τ .Up until the strongest flow amplitude, the large forces cannot disrupt the synchronized flagellar beating.In addition, synchronization is never established around frequencies other than f 0 .This shows that the inter-flagella coupling is much stronger than the maximum amplitude of forcing.
Next we examine the synchronization with the θ c -flows and θ t -flows.Flows of a fixed amplitude (∼ 7U 0 ) but varying frequencies around f 0 are applied to each captured cell (see Methods).With these, the flow-synchronized time fraction τ as a function of the detuning (ν) and flow direction (θ c,a,t ) is recorded and helps quantify the flows' effective forcing ε(θ).
Comparing τ (ν; θ c ) to τ (ν; θ t ), with τ (ν; θ a ) as reference, we find that θ c -flows are the most effective in synchronizing the beating (Fig. 3B).We illustrate this point with the profiles of an exemplary cell (Fig. 3B inset).First, although both the θ c -flow (red) and the θ t -flow (blue) can synchronize the cell at small detunings (|ν| <0.5Hz), the θ c -flow maintains the synchronization for the whole time ( τ (θ c ) =1), while the θ t -flow for a slightly smaller time fraction ( τ (θ c ) ≈0.85).This is due to phase-slips (step-like changes in ∆ϕ(t) in Fig. 1F) between flagella and the flow, and means that the θ t -flow synchronization is less stable.Additionally, for intermediate detuning (0.5 Hz< |ν| <4 Hz), τ (θ c ) is always larger than τ (θ t ) .In some cases, the θ c -flow synchronizes the cell fully whereas the θ t -flow fails completely (e.g., at ν = −2 Hz).Together, these results imply that a flow of given amplitude synchronizes flagellar beating more effectively if it selectively loads the cis.
We repeat the experiments with cells from multiple cultures, captured on different pipettes, and with different eyespot orientations (∼50% heading rightward in the imaging plane) to rule out possible influence from the setup.τ (ν; θ) of N=11 wt cells tested in the TRIS-minimal medium (pH=7.0)are displayed in Fig. 3B (labeled as "TRIS").On average, ε(θ c ) = 2.9 Hz and is 70% larger than ε(θ t ) = 1.7 Hz.It bears emphasis that ε(θ c ) > ε(θ t ) holds true for every single cell tested (11/11).In Fig. 3C, we show this by representing each cell as a point in the for this cell.All cells cluster clearly below the line.This asymmetry manifest equivalently through τ .In Fig. 3D, each point represents the time fractions of the same cell synchronized by the θ c -flow and the θ t -flow at the same frequency.Most points (>90%) are below the first bisector line, meaning that τ (θ c ) > τ (θ t ) .Altogether, all results show that selectively loading the cis flagellum establishes synchronization with the flow more effectively, pointing to cis and trans playing differential roles in the coordinated beating.
We next study whether this newly observed cis-trans asymmetry is affected by calcium depletion.Calcium is a critical second messenger for modulating flagellates motility and is deeply involved in phototaxis (45).The depletion of the free environmental calcium is known to degrade flagellar synchronization and exacerbate flagellar asymmetry (22).Here we focus on whether calcium depletion affects the asymmetry ε(θ c ) > ε(θ t ) .We deplete environmental calcium by EGTA-chelation, following the protocol in Ref. (46).Similar to previous reports (22,47), the number of freely swimming cells drops significantly in EGTA-containing medium.
However, the remaining cells beat synchronously for hours after capture.For these beating cells, calcium depletion is first confirmed by characterizing their deflagellation behavior.Indeed, calcium depletion is reported to inhibit deflagellation (28,48).In experiments with standard calcium concentration, all cells deflagellated under pipette suction (20/20).For experiments conducted in calcium depleting EGTA-containing medium, we observe deflagellation to occur in none but one cell (1/19).
After confirming the calcium depletion, we perform the same sets of flow synchronization experiments.The dashed lines in Fig. 3B show the median synchronization profiles τ (ν; θ) (N=6 cells, labeled as "EGTA").The flagellar asymmetry is unaffected, see also Fig. 3E.Note that ε(θ c ) > ε(θ t ) again applies for every single cell tested.The mean values of ε drop slightly.However, the different effectiveness between θ c -flows and Finally, we determine how the forcing strength of the flow depends on the hydrodynamic forces exerted by the flow on the flagella.We compute the hydrodynamic beat-averaged loads, ) for these flows remains almost constant (0.74-0.79F 0 ), the zero-intercept implies that for a hypothetical flow that exerts no load on the cis but solely forces the trans, it will not be able to synchronize the cell at all.This suggests a negligible contribution of the forcing on the trans in establishing synchronization with flows.

The asymmetry is lost in ptx1 mutants
Furthermore, we examine the flagellar dominance mutant ptx1.In this mutant, both flagella respond similarly to changes of calcium concentrations (38) and have similar beating frequencies when demembranated and reactivated (23).
Ptx1 mutants have two modes of coordinated beating, namely, the in-phase (IP) synchronization and the anti-phase (AP) synchronization (29,49).First, we apply θ a -flow in the same frequency and amplitude ranges as for wt cells.We find that the IP mode around f 0 ≈ 50 Hz is the only mode that can be synchronized by external flows.We focus on this mode and report τ as τ = t sync /t IP for this mutant, where t IP is the total time of IP-beating under the applied flows, see Fig. 4A.Synchronization profiles τ (ν; θ) of ptx1 are shown in Fig. 4B.The median profiles are of similar width and height, indistinguishable from each other, and hence indicate a loss of asymmetric susceptibility to flow synchronization.The loss is further confirmed by the extracted ε(θ) (31) and τ (θ) (Fig. 4C-D).Cells and synchronization attempts are distributed evenly across the first bisector lines (7/14 cells are below ε(θ c ) = ε(θ t ) in Fig. 4C, and ∼50% points are below τ (θ c ) = τ (θ t ) in Fig. 4D).Altogether, all results show consistently that the asymmetry is lost in ptx1.

Modeling Framework
To investigate the implications of our experimental results on the coupling between flagella and their dynamics, we develop a model for the system (SM.Sec.S3), representing flagella and external flows as oscillators with directional couplings: ϕ f,c,t (t) respectively represent the phase of the flow, the cis, and the trans flagellum.f f,c,t represents the inherent frequency of the forcing (flow), the cis, and the trans respectively.The phase dynamics of each flagellum is modulated by its interactions with the other flagellum as well as the background flow.Take the cis ( φc ) for example, the effect of the trans and the forcing on the cis are respectively accounted for by the λ t -term and the ε c -term, see Eq. ( 2).In other words, λ t and ε c measure the sensitivity of the actual cis-frequency to the phase differences between oscillators (ϕ c − ϕ t,f ), see the arrows in Fig. 5A.Lastly, ζ c,t represent the white noise of the cis and trans flagellum respectively.In the following parts, without loss of generality, the noise are assumed equally strong and uncorrelated ( . Nuanced phase dynamics under differential noise levels can be found in SM.Sec.S4. Eq. ( 2) can be readily reduced to Eq. ( 1), which allows us to write the experimentally measured values (f 0 , ε(θ), T eff ) analytically with ε c,t , λ c,t , and ζ c,t .The asymptotic behavior of the model under the condition ϕ c ≈ ϕ t ≈ ϕ f are (SM.Sec.S3): with α = λ c /(λ c + λ t ) representing the dominance of cis.It is then clear that when α ≈ 1, the coordinated beating will display dynamic properties of the cis flagellum.respectively.The selective loading on the cis is represented by ε c > ε t ; while λ c > λ t reflects that the cis has a more dominant role in the coordinated beating.We run Monte-Carlo simulation with Eq. ( 2) using customized MATLAB scripts.
At similar detunings as in the experimental results in Fig. 1F, our Monte-Carlo simulations reproduces the phase dynamics with: (i) no flow synchronization, (ii-iii) unstable synchronization, and (iv) stable synchronization (Fig. 5B).Repeating the simulations for varying forcing strength ε (= ε c,t ) and frequency f f yields Arnold tongue diagrams in agreement with those reported from our experiments.The Arnold Tongue for wt in Fig. 3A and ptx1 in Fig. 4A are reproduced with simulations shown in Fig. 5C and D respectively.The only parameter value changed between Fig. 5C and D is the level of noise (T c,t eff ), which is increased by an order of magnitude.The differences in phase dynamics between wt and ptx1, when subjected to symmetric external loading, are therefore accounted by solely varying the noise.

Coordinated beating under selective loading
We next model flow synchronization by the θ c -flows and the θ t -flows.The selective forcing (ε c = ε t ) allows the effect of flagellar dominance (λ c = λ t ) to manifest in the effective forcing strength ε(θ) and hence in the synchronization profiles τ (ν; θ), Fig. 5E.Similar to our experimental observations, θ c -flow synchronizes the coordinated beating over the broadest range of ν (i.e. largest ε).This is directly attributed to the dominance λ c > λ t : by setting λ c = λ t , the differences between τ (θ c ) and τ (θ t ) disappear even under selective loading (Fig. 5E inset).3).The difference between ε(θ c ) and ε(θ t ) increases with λ c /λ t , and they each saturates to reflect only the forcing on the cis (ε c , the grey dashed lines).With f c = 45 Hz, f t = 65 Hz (23,26), and f 0 ≈ 50 Hz, we deduce from Eq. ( 3) that λ c = 4λ t for wt cells.For wt cells under calcium depletion, experimental results are reproduced with a lower total forcing strength (Fig. 5G).ε c + ε t is set to 4.08 Hz (15% lower) to reflect the 7% − 20% decrease in ε(θ) induced by calcium depletion.
The ptx1 results are reproduced with a stronger noise (T c,t eff = 9.42 rad 2 /s) and a symmetric inter-flagellar coupling λ c /λ t = 1, see Fig. 5H and Table. 1.Both changes are necessary for reproducing the synchronization profiles of ptx1 in Fig. 5H: while the stronger noise lowers the maximal values of τ (θ, ν), setting λ c /λ t = 4 would still result in τ (θ c ) > τ (θ t ) in the central range (|ν| 2.4 Hz).Finally, it is noteworthy that the noise in ptx1 increases not only because a higher noise value for individual flagella, but also because the cis-trans coupling has become symmetric.As shown by Eq. ( 3), the unilateral coupling promotes not only the cisfrequency in the synchrony but also the cis-noise.Given T c eff T t eff and λ c = 4λ t , we confirm with simulations that the cis stabilizes the beating frequency of the trans and decreases its beating noise.The simulations are in good agreement with experimental noise measurements, see SM. Sec.S4 for details.

Discussion
The two flagella of C. reinhardtii have long been known to have inherently different dynamic properties such as frequency, waveform, level of active noise, and responses to second messengers (23,25,26,29,30).Intriguingly, when connected by basal fibers and beating synchronously, they both adopt the kinematics of the cis-(eyespot) flagellum, which led to the assumption that the flagella may have differential roles in coordination.In this work, we test this hypothesis by employing oscillatory flows applied from an angle with respect to the cells' symmetry axis and thus exert biased loads on one flagellum.
Without an exception, in wt cells, θ c -flows, the ones that selectively load the cis flagellum, are always more effective in synchronizing the flagellar beating than the θ t -flows.This is shown by the larger effective forcing strengths ( ε(θ c ) > ε(θ t ) , Fig. 3B-C) and larger synchronized time fractions ( τ (θ c ) > τ (θ t ) , Fig. 3D).Mapping the measured forcing strength ε(θ) as a function of the loads, we find empirically that ε ∝ F c Flow (Fig. 3F) and that trans-loads appear to matter negligibly.These observations all indicate that the cis-loads determine whether an external forcing can synchronize the cell.Moreover, this point is further highlighted by an unexpected finding: when θ t -flows are applied, the trans flagellum always beats against the external flow (P t Flow < 0) and the only stabilizing factor for flow synchronization is the cis flagellum working along with the flow during the recovery stroke (Fig. 2C lower panel).These observations definitively prove that the two flagella have differential roles in the coordination and interestingly imply that flagella are coupled to external flow only through the cis.
To have a mechanistic understanding of this finding, we model the system with Eq. ( 2).In the model, selective hydrodynamic loading and flagellar dominance in the coordinated beating are respectively represented by ε c = ε t and λ c = λ t .Setting out from the model, we obtain closed-form expressions for observables such as f 0 and ε (Eq.( 3)), which illustrate how flagellar dominance and selective loading affect the coordinated flagellar beating.Moreover, with Monte-Carlo simulation, we clarified the interplay between flows and flagella (SM.Sec.S3), and reproduces all experimental observations.With the model, we show that a "dominance" of the cis (λ c > λ t ) is sufficient to explain why the coordinated flagellar beating bears the frequency and the noise level of the cis flagellum.In the model, such dominance means that the cis-phase is much less sensitive to the trans-phase than the other way around.We then reproduce the phase dynamics of flow synchronization at varying detunings (Fig. 5B), amplitudes (Fig. 5C), and noise (Fig. 5D).Exploiting the observation that the coordination between flagella cannot be broken by external flows up to the strongest ones tested (ε max ∼ 10 Hz, Fig. 3A), we quantify the lower limit of the total basal coupling, λ c + λ t , to be approximately 40 Hz (deduced in SM.Sec.S3), which is an order magnitude larger than the hydrodynamic inter-flagellar coupling (31,(50)(51)(52).
The modulation of flagellar dominance mediates tactic behaviors (22,23,38,47).Calcium is hypothesized to be underlying the modulation of dominance, as it causes the connecting fiber between flagella to contract (53), modulates the cis-and trans activity (e.g.beating amplitude) differentially (22), and calcium influx comprises the initial step of CR's photo- (54) and mechanoresponses (45).We therefore investigate flagellar coupling in the context of tactic steering by depleting the environmental free calcium and hence inhibiting signals of calcium influxes.Cells are first acclimated to calcium depletion, and then tested with the directional flows.Our results show that the cis dominance does not require the involvement of free envi-ronmental calcium.Calcium depletion merely induces an overall drop in the forcing strength perceived by the cell ε(θ) (7% − 20%), which is captured by reducing ε c + ε t for 15% (mean drop) in the model (Fig. 5G).Together, our results indicate that the leading role of cis, is an inherent property, that does not require active influx of external calcium, and possibly reflects an intrinsic mechanical asymmetry of the cellular mesh that anchors the two flagella into the cell body.
In ptx1 cells, a lack of flagellar dominance (λ c = λ t ) and a stronger noise level help reproduce our experimental observations.Previous studies suggested that both flagella of ptx1 are similar to the wildtype trans (23), and that the noise levels of this mutant's synchronous beating are much greater than those of wt (29) (see also SM.Sec.S4).If both flagella and their anchoring roots indeed have the composition of the wildtype trans, such symmetry would predict λ c = λ t .
This symmetric coupling renders the noise of ptx1 T eff = T t eff (Eq.( 3)), which is about an order of magnitude larger than the noise of wt T eff ≈ T c eff .The comparison between ptx1 and wt highlights an intriguing advantage of the observed unilateral coupling (λ c λ t ); that is, it strongly suppresses the high noise of the trans.Considering that the trans is richer in CAH6 protein and this protein's possible role in inorganic carbon sensing (14,20), the potential sensing role of the trans is worth noticing.Assuming the strong noise present in the trans originates from the biochemical processes related to sensing, then the unilateral coupling effectively prevents such noise from perturbing the cell's synchronous beating and effective swimming.In this way, the asymmetric coupling may combine the benefit of having a stable cis as the driver while equipping a noisy trans as a sensor.

Cell culture
CR wildtype (wt) strain cc125 (mt+) and flagellar dominance mutant ptx1 cc2894 (mt+) are cultured in TRIS-minimal medium (pH=7.0)with sterile air bubbling, in a 14h/10h day-night cycle.Experiments are performed on the 4th day after inoculating the liquid culture, when the culture is still in the exponential growth phase and has a concentration of ∼ 2 × 10 5 cells/ml.Before experiments, cells are collected and resuspended in fresh TRIS-minimal (pH=7.0).

Calcium depletion
In calcium depletion assays, cells are cultured in the same fashion as mentioned above but washed and resuspended in fresh TRIS-minimal medium + 0.5 mM EGTA (pH=7.0).Free calcium concentration is estimated to drop from 0.33 mM in the TRIS-minimal medium, to 0.01 µM in the altered medium (46).Experiments start at least one hour after the resuspension in order to acclimate the cells.

Experimental setup
Single cells of CR are studied following a protocol similar to the one described in (31).Cell suspensions are filled into a customized flow chamber with an opening on one side.The airwater interface on that side is pinned on all edges and is sealed with silicone oil.A micropipette held by micromanipulator (SYS-HS6, WPI) enters the chamber and captures single cells by aspiration.The manipulator and the captured cell remain stationary in the lab frame of reference, while the flow chamber and the fluid therein are oscillated by a piezoelectric stage (Nano-Drive, Mad City Labs), such that external flows are applied to the cell.Frequencies and amplitudes of the oscillations are individually calibrated by tracking micro-beads in the chamber.Bright field microscopy is performed on an inverted microscope (Nikon Eclipse Ti-U, 60× water immersion objective).Videos are recorded with a sCMOS camera (LaVision PCO.edge) at 600-1000 Hz.

Measurement scheme
The flagellar beating of each tested cell is recorded before, during, and after the application of the flows.We measure the cell's average beating frequency f 0 over 2 s (∼100 beats).For ptx1 cells, f 0 is reported for the in-phase (IP) synchronous beating.Unless otherwise stated, directional flows (θ = 0, ±45 • ) are of the same amplitude (780±50 µm/s, mean±std), similar to those used in Ref. (31).Flow frequencies f f are scanned over [f 0 − 7, f 0 + 7] Hz for each group of directional flows.

Computation of the flagellar loads
To quantify the hydrodynamic forces on the flagella, we first track realistic flagellar deformation from videos wherein background flows are applied.Then we employ a hybrid method combining boundary element method (BEM) and slender-body theory (40,55) to compute the drag forces exerted on each flagellum and the forces' rates of work.In this approach, each flagellum is represented as a slender-body (55) with 26 discrete points along its centerline and the time-dependent velocity of each of the 26 points is calculated by its displacement across frames.
The cell body and the pipette used to capture the cell are represented as one entity with a completed double layer boundary integral equation (56).Stresslet are distributed on cell-pipette's surface; while stokeslet and rotlet of the completion flow are distributed along cell-pipette's centerline (57).The no-slip boundary condition on the cell-pipette surface is satisfied at collocation points.Lastly, stokeslets are distributed along the centerlines of the flagella, so that no-slip boundary conditions are met on their surfaces.Integrating the distribution of stokeslets f(s) over a flagellar shape, one obtains the total drag force F = f(s)ds is obtained.Similarly, the force's rate of work is computed as P = f(s) • U(s)ds, where U(s) is the velocity of the flagellum at the position s along the centerline.
The computations shown in this study are based on videos of a representative cell which originally beats at ∼50 Hz.The cell is fully synchronized by flows along different directions (θ = 0 • , ±45 • and 90 • ) at 49.2 Hz.In the computations, the applied flows are set to have an amplitude of 780 µm/s to reflect the experiments.Computations begin with the onset of the background flows (notified experimentally by a flashlight event), and last for ∼30 beats (500 frames sampled at 801 fps).Additionally, we confirm the results of θ t -flow-synchronization, that both flagella spend large fractions of time beating against the flows, with other cells and with θ t -flows at other frequencies.

Isolate loads of external flows
The total loads (F and P ) computed consist of two parts, one from the flow created by the two flagella themselves and the other from the applied flow.In the low Reynolds number regime, the loads of the two parts add up directly (linearity): F = F Self + F Flow , and P = P Self + P Flow .
To isolate F Flow and P Flow , we compute F = F Self and P = P Self by running the computation again but without the external flows, and obtain F Flow = F − F and P Flow = P − P .

Modeling parameters
We assume the flagellar intrinsic frequencies f c and f t to be 45 Hz and 65 Hz respectively (23,26,28).On this basis, λ c : λ t is assumed to be 4:1 to account for the observed f 0 (∼ 50 Hz).ε c : ε t is set as 2:1, 1:1, and 1:2 for the θ c -flows, the θ a -flows, and the θ t -flows respectively, see Fig. 2A-C.Additionally, ε c + ε t is assumed to be constant to reflect the fact that F c Flow + F t Flow approximately does not vary with flow directions.We take a typical value of T c,t eff = 1.57rad 2 /s (31).The sum of inter-flagellar coupling λ tot = λ c + λ t is set to be large enough, i.e., λ tot = 3ν ct with ν ct = |f t − f c |, to account for the fact that: 1) the coordinated beating is approximated in-phase, and 2) up until the strongest flow applied, the coordinated beating cannot be broken (quantitative evaluation is detailed in SM.Sec.S3).To model wt cells under calcium depletion, we decrease ε c + ε t by 15% -which is the mean decrease in the observed ε(θ c ) , ε(θ a ) , and ε(θ t ) (Fig. 3E).For ptx1 cells, we assume a symmetric inter-flagellar coupling (λ c = λ t ) and a stronger noise level (SM.Sec.S4).The parameters are summarized in Table .1.

S3.3 Lower limit of inter-flagellar coupling
The value (λ tot − ν ct )/ε determines if the flow can disrupt the synchronization between cis and trans.We assume ν ct = 20 Hz [4,5,6,3] and focus on synchronization of the θ a -flow.
We plot the synchronization time fractions with increasing λ tot in Fig. S3.When it satisfies (λ tot − ν ct )/ε ≥ 2, external flows cease to affect the flagellar synchronization observably.As the strongest flow (21U 0 ) applied experimentally corresponds to ε ≈ 10 Hz, altogether, we conclude that λ tot ν ct + 2ε max = 40 Hz.In the main text, we set λ tot = 60 = 3ν ct Hz, which satisfies this relation and matches the observation that the phase lag between the flagella (δ ct ) is small.

S4 Flagellar noise of the ptx1 mutant
Here we show an as-yet uncharacterized strong noise present in the synchronous beating of the mutant ptx1.The in-phase (IP) mode of ptx1 cells and the breaststroke beating of the wt cells are similar in waveform and frequency [7,8].However, the former has a much stronger noise.The strong noises show obviously in fluctuations of IP beating frequencies [8].
In Fig. S4, we display the distribution of beating frequency of a representative wt cell (panel a) and four representative ptx1 cells (panels b-e).The broad peaks of the IP (purple) and AP (yellow) beating of ptx1 sharply contrast the narrow peak of wt.We quantify the frequency fluctuations of all the cells in the main text (N=11 for wt and N=14 for ptx1), Fig. S4f.The cells are represented by its mean beating frequency over time f 0 and the frequency's standard deviation σ(f 0 ).Clearly, the breaststroke beating of wt, the IP, and the AP mode of ptx1 each forms a cluster.The wt cluster is at ( f 0 , σ(f 0 )) = (50.5 ± 2.6, 0.8 ± 0.3) Hz (mean± 1 std. the over cell population); and it is evidently less dispersed than both the IP and the AP mode of ptx1, which are at (47.4 ± 3.1, 3.4 ± 0.9) Hz and (67.6 ± 2.1, 1.9 ± 0.7) Hz, respectively.
Under the assumption of a white (Gaussian) noise, σ(f 0 ) is proportional to the noise level ζ, and thus scales with √ T eff .Consider that σ(f 0 ) for ptx1 is 3-5 folds larger than that of wt, we therefore conclude that the noise level in ptx1 is an order of magnitude larger than wt, T ptx1 eff /T wt eff ∼ O (10).The stronger noise in ptx1 can be attributed to two sources, namely, the loss of a stable cis and the loss of the unilateral coupling, Fig. S5.We perform Monte-Carlo simulations of the coupled beating of cis and trans under three conditions: (1) a stable cis (T c eff = T 0 eff = 1.57rad/s 2 ) coupled with the trans unilaterally (λ c = 4λ t ), (2) a stable cis coupled with the trans bilaterally (λ c = λ t ), and (3) an equally noisy cis (T c eff = T t eff ) bilaterally coupled with trans, see the blue, yellow, and red data in Fig. S5 respectively.It is obvious that, when the trans is coupled to a stable cis, varying its noise over an order of magnitude only leads to a ∼ 20% stronger frequency fluctuation (the blue line in Fig. S5(a)).On the contrary, lacking either the unilateral coupling or the low-noised cis would increase the fluctuation for 200% (yellow line) or 300% (red line).Qualitatively, simulation results are in agreement with experimental measurements assuming that T t eff /T c eff ∼ O (10), see the red and blue shaded areas in Fig. S5(a).Moreover, a low-noise cis is already sufficient to prevent slips from interrupting the synchrony between cis and trans, even for bilateral coupling.In Fig. S5(b), as long as the cis-noise remains low, slips will be sparse (< 0.01 Hz).Together, these simulation results highlight the stabilizing effect of a low-noise cis flagellum, and illustrates the contribution of unilateral coupling in further enhancing the stabilization.
al. (31), we induce oscillatory background flows to exert hydrodynamic forcing to flagella of captured cells.With programmed oscillations of the piezoelectric stage, the amplitude, frequency, and direction of the background flows are all controlled, enabling selective loading.To quantitatively estimate the selectivity of the flows along different angles (θ), we compute the flagellar loads under the flows along θ = −45 • , 0 • , and 45 • , see Fig. 1A.Computations based on boundary element methods (BEM) and slender-body theory (SBT) give the real-time drag force F on each flagellum and the power P exerted by the viscous forces on each flagellum.
2π, P Flow = 2π 0 P Flow dϕ/2π, induced by the flow on the trans and on the cis flagella, see the horizontal lines in Fig. 2.These loads are computed for the θ c -flow, θ t -flow, θ a -flow and we also include experiments and computations performed with flows along θ = 90 • (circles), see SM. Sec.S2.Fig. 3F and G represent ε as a function of the loads on the cis and trans flagellum respectively, with each symbol representing one of the four different flow directions, see the drawings.We find that the effective forcing strength scales with the time-averaged drag on the cis, ε ∼ F c Flow , while we find no such correlation between ε and F t Flow .The linear relation between ε and F c Flow has an intercept near zero (ε| F c Flow =0 ≈ 0).Given the total forces on both flagella (F c Flow + F t Flow

Fig
Fig. 5A illustrates an exemplary modeling scheme describing flagellar beating subjected to θ c -flows.The direction and thickness of arrows represent coupling direction and strength

Figure 1 :
Figure 1: Experimental workflow.(A) Captured CR cells are subjected to sinusoidal flows of frequency f f along given angles (θ) in the xy-plane.Flows along θ = −45 • , 0 • , 45 • of same amplitude (780±50 µm/s, mean±std.)are used and termed as shown.(B-E) Extracting flagellar phase ϕ c and ϕ t by image processing.Raw images (B) are thresholded and contrast-adjusted to highlight the flagella (C).Mean pixel values within the user-defined interrogation windows (red and blue circles) capture the raw phases of beating (D), which are then converted to observableindependent phases (E).Inset: phase difference ϕ c − ϕ t .(F) Flagella-flow phase dynamics at decreasing detuning ν = f f − f 0 with f 0 the cell's beating frequency without external flow.Traces i to iv are taken at detunings marked in the inset.Plateaus marked black represent flow synchronization, whose time fractions τ = t sync /t tot are noted.t tot is the total time of recording.Inset: the flow synchronization profile, τ (ν), reports the effective forcing strength 2ε by its width.

Figure 3 :
Figure 3: Flow synchronization of wt cells.(A) Arnold tongue of a representative cell tested with θ a -flow.The contour is interpolated from N=132 measurements (6 equidistant amplitudes × 22 equidistant frequencies), and color-coded by the entrained time fraction τ .(B) The synchronization profiles τ (ν; θ) of a representative wt cell (inset), the median profile of the TRIS group wt cells (N=11, solid lines) and the EGTA group (N=6, dashed lines), with either θ cflows (red), θ a -flows (yellow) or θ t -flows (blue).Shaded areas are the interquartile ranges for the TRIS group.(C) Tested wt cells represented on the ε(θ c ) − ε(θ t ) plane (TRIS group).Solid line: the first bisector line (y = x).(D) Comparing τ (ν; θ c ) and τ (ν; θ t ) for each cell at each applied frequency.N=132 pairs of experiments are represented on the τ (θ c ) − τ (θ t ) plane.More than 90% of them are below the first bisector line.(E) The coupling strengths ε(θ) of the TRIS group (black) and the EGTA group (gray).Bars and error bars: mean and 1 std., respectively.Inset: δε = ε(θ c ) − ε(θ t ) .NS: not significant, p>0.05,Kruskal-Wallis test, One-Way ANOVA.Relations between the forcing strength ε and the loads on the cis (F) and the trans flagellum (G).Markers represent different flow angles, see the drawings.

Figure 5 :
Figure 5: Modeling the asymmetric flow synchronization.(A) Modeling scheme describing a cell beating under directional flow (θ c -flow as an example).Arrows represent the directional coupling coefficients with line thickness representing the relative strength.For example, λ c points from cis to trans, representing how the latter (ϕ c ) is sensitive to the former (ϕ t ); meanwhile, the arrow of λ c being thicker than λ t means that ϕ t is much more sensitive to ϕ c than the other way around.(B) Modeled phase dynamics of flow synchronization under θ a -flows, analogous to Fig. 1F.Reproducing the Arnold tongue diagrams at the noise level of wt (C) and ptx1 (D), analogous to Fig. 3A and Fig. 4A respectively.(E) Flow synchronization profiles τ (ν; θ) obtained experimentally (upper panel) and by modeling (lower panel).Inset: the modeling results with symmetric inter-flagellar coupling.(F) Effective forcing strength ε(θ) as a function of the inter-flagellar coupling asymmetry λ c /λ t .Points: measured from simulation; lines: analytical approximation (Eq.(3)); dashed lines: ε c respectively for the θ c -flow, θ a -flow, and θ t -flow (from top to bottom).(G) Reproducing the flow synchronization of wt cells under calcium depletion (H) Reproducing results of ptx1.See Table. 1 for the modeling parameters.

Figure S1 :
FigureS1: Equivalence of extracting coupling strength ε by different methods.Each point represents one cell under either the θ a -flow (green square), the θ c -flow (red circle), or the θ tflow (blue triangle).The x coordinate is the coupling strength ε measured by the half width of synchronization profile τ (ν) ≥ 50%; and the y coordinate is obtained by fitting the flagellar phase dynamics.

Figure S2 :Figure S3 :
Figure S2: Computed hydrodynamic loads on the flagella.Computation results of the drag force (upper panel) and the force's rate of work (lower panel) on the cis (red) and the trans (blue) flagellum during synchronized cycles, when the cell is subjected to the flow with θ = 90 • .Scaling factors F 0 =9.9 pN and P 0 =1.1 fW.

Figure S4 :
Figure S4: Stronger frequency fluctuation of the IP mode of ptx1 cells.(a-e) Representative probability distributions of the beating frequency of a wt (a) and four ptx1 cells (b-e) over 30 seconds.Probability distributions of the IP (purple) and AP mode (yellow) are respectively normalized for better visualization.The time fractions of the AP mode are noted in each panel.(f) The wt and ptx1 cells represented by its mean beating frequency f 0 and the standard deviation of the beating frequencies over time σ(f 0 ).

Figure S5 :
Figure S5: Effect of a low-noise cis in stabilizing the beating of the trans (a) Fluctuations in beating frequency (σ(f 0 )) under different coupling schemes and flagellar noises.Other model parameters are the same as used in the main text.The red and blue shaded area represent the experimentally observed range for ptx1 and wt cells, respectively, with short bars marking the mean values.(b) the rate of slip under the conditions.Error bars correspond to 1 std.over N=9 repetitions.